Full Isometric Grouped

In Sect. 2.2.2 we have shown that if a SRM admits primitive period isometric transformations, representations of two groups and may be derived. Extension of a representation of J7 by leads to the corresponding representation of St, whereas extension of the representations of St by g gives those of St. The use of St or depends on the problem to which the isometric group is to be applied, as has been pointed out in Section 2.2.2. In order to simplify the notation we shall for general discussions not distinguish between the representations of St and St. 

Therefore, the internal isometric group generates on the position vectors of the nuclei a set of o I IJH matrices 

The set of solutions (2.70) defines the representation r Ncf of the full isometric group F( ) on the nuclear position vectors referred to the frame system 

The decomposition of r(NCf) JF modulo the invariant subgroup r(NCf) g defines a factor group isomorphic to the internal isometric groups (I) 

As will be shown below [Eq. (2.80)] there exists always a subgroup r(NCf) in r(NCf SP isomorphic to this factor group. Therefore, Eq. (2.74) suggests the following important theorem  

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The group (2.49") will prove important for the construction of the full isometric group, cf. Sect. 2.2.4. 

Analogously for SRMs with proper covering group g) and primitive period transformations, the full isometric group Wig) =Jrg) ( ) is homomorphic to the permutation-inversion group... 

Table 13c. Generators of the full isometric group sr (t) of the SRM D, , F(C2vT)2...

T full isometric group in our example is isomorphic to a group of order... 

Again this relation follows from the symmetries (ii) and (iii) it expresses that the electronic energy function assumes the same value for all isometric NCs. Equations (3.5) and (3.9) show that e°(Xk( )) is symmetric w.r.t. to the full isometric group. Whereas the symmetry of e°(Xk( )) w.r.t. ( ) merely expresses that e°(Xk( )) is a function of the internal coordinates only, its symmetry w.r.t. Jr( ) is a genuine symmetry. 

The foregoing discussion allows to state the theorem The full isometric group JP( ) is a proper or improper subgroup ofthe symmetry group ft of the rotation internal motion hamiltonian H = T + V... 

One of the difficulties of the NRG theory is to construct correctly the full, as well as the restricted Hamiltonian operators, and to deduce properly the physical operations which commute with these operators. In many cases, however, the interconversion operations may be described easily as rotations of molecular moieties around some axes supported by a solid molecular frame. In such cases, the concept of restricted NRG recovers special relevance, and the restricted NRG appears to be equivalent to the isometric group [27]. 

The concepts of full and restricted NRG s are defined. The full NRG s, which consider the overall rotations, are seen to be entirely equivalent to the Molecular Symmetry Groups of Longuet-Higgins [5], whereas the restricted NRG s, limited to the interconversion motions, may be compared with the Isodynamic groups of Altmann, and the isometric groups of Gunthard [10,27]. 

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